Term Rewriting System R:
[x, y, z, x1, x2, x3, x4]
f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F1 -> G1
F1 -> G2
F2 -> G1
F2 -> G2
E1(h1, h2, x, y, z) -> E2(x, x, y, z, z)
E2(f1, x, y, z, f2) -> E3(x, y, x, y, y, z, y, z, x, y, z)
E3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> E4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
E4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> E1(x1, x1, x, y, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

E4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> E1(x1, x1, x, y, z)
E3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> E4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
E2(f1, x, y, z, f2) -> E3(x, y, x, y, y, z, y, z, x, y, z)
E1(h1, h2, x, y, z) -> E2(x, x, y, z, z)


Rules:


f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

E1(h1, h2, x, y, z) -> E2(x, x, y, z, z)
one new Dependency Pair is created:

E1(h1, h2, f1, y'', f2) -> E2(f1, f1, y'', f2, f2)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

E3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> E4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
E2(f1, x, y, z, f2) -> E3(x, y, x, y, y, z, y, z, x, y, z)
E1(h1, h2, f1, y'', f2) -> E2(f1, f1, y'', f2, f2)
E4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> E1(x1, x1, x, y, z)


Rules:


f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

E4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> E1(x1, x1, x, y, z)
one new Dependency Pair is created:

E4(g1, x1', g2, x1', g1, x1', g2, x1', f1, y', f2) -> E1(x1', x1', f1, y', f2)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pairs:

E2(f1, x, y, z, f2) -> E3(x, y, x, y, y, z, y, z, x, y, z)
E1(h1, h2, f1, y'', f2) -> E2(f1, f1, y'', f2, f2)
E4(g1, x1', g2, x1', g1, x1', g2, x1', f1, y', f2) -> E1(x1', x1', f1, y', f2)
E3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> E4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)


Rules:


f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

E3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> E4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
one new Dependency Pair is created:

E3(x1', x1', x2', x2', x3', x3', x4', x4', f1, y', f2) -> E4(x1', x1', x2', x2', x3', x3', x4', x4', f1, y', f2)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

E1(h1, h2, f1, y'', f2) -> E2(f1, f1, y'', f2, f2)
E4(g1, x1', g2, x1', g1, x1', g2, x1', f1, y', f2) -> E1(x1', x1', f1, y', f2)
E3(x1', x1', x2', x2', x3', x3', x4', x4', f1, y', f2) -> E4(x1', x1', x2', x2', x3', x3', x4', x4', f1, y', f2)
E2(f1, x, y, z, f2) -> E3(x, y, x, y, y, z, y, z, x, y, z)


Rules:


f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)





On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

E2(f1, x, y, z, f2) -> E3(x, y, x, y, y, z, y, z, x, y, z)
one new Dependency Pair is created:

E2(f1, f1, y', f2, f2) -> E3(f1, y', f1, y', y', f2, y', f2, f1, y', f2)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

E4(g1, x1', g2, x1', g1, x1', g2, x1', f1, y', f2) -> E1(x1', x1', f1, y', f2)
E3(x1', x1', x2', x2', x3', x3', x4', x4', f1, y', f2) -> E4(x1', x1', x2', x2', x3', x3', x4', x4', f1, y', f2)
E2(f1, f1, y', f2, f2) -> E3(f1, y', f1, y', y', f2, y', f2, f1, y', f2)
E1(h1, h2, f1, y'', f2) -> E2(f1, f1, y'', f2, f2)


Rules:


f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)




Termination of R could not be shown.
Duration:
0:34 minutes